Two-phase flow simulation in low-permeability heterogeneous reservoirs with a fully implicit scheme-based enriched physics-informed neural network

Two-phase flow simulation in low-permeability heterogeneous reservoirs is challenging because low-velocity non-Darcy flow, capillary pressure, fractures, and heterogeneity produce tightly coupled and highly nonlinear pressure–saturation equations. A Fully Implicit Scheme-based Enriched Physics-Informed Neural Network (EPINN-FIS) is developed to solve this problem without labeled simulation data. The governing equations are discretized using the finite volume method, and the resulting oil- and water-phase mass-balance residuals are used to train a coupled pressure–saturation network. Unlike physics-informed formulations that rely on implicit-pressure-explicit-saturation updates, EPINN-FIS evaluates pressure, saturation, phase mobilities, non-Darcy coefficients, and capillary terms simultaneously at the new time level. Adjacency-location anchoring, adaptive activation functions, skip connections, gated updating, and parameter transfer between consecutive time steps are incorporated to improve training robustness. Numerical tests include two-dimensional heterogeneous and fractured reservoirs and a three-dimensional corner-point-grid model. The predicted pressure and saturation fields and well responses agree closely with the reference fully implicit simulator. Across the tested cases, pressure relative errors remain below approximately 1%, while saturation absolute errors remain below 0.02; the maximum saturation error in the three-dimensional case is below 0.004. EPINN-FIS therefore provides a physically constrained, Jacobian-free alternative solution framework for strongly nonlinear two-phase-flow problems. Although the present implementation is computationally more expensive than an optimized conventional simulator, improving its scalability remains an important direction for future work.